# Math Help - Smallest Positive integer n

1. ## Smallest Positive integer n

Find the form of all n in N satifying tow(n)=6. (sorry don't know how to write this Tex). What is the smallest positive integer n for which this is true?

tow(n)=6 so 6=(1+a1)(1+a2)----(1+ak) where n=p1^a1p2^a2---pk^ak

2. $\tau(n)=6$
We have $\tau(n)=(1+\alpha_{1})(1+\alpha_{2}).....(1+\alpha _{k})$for k= $p^{\alpha_{1}}p^{\alpha_{2}}...p^{\alpha_{k}}$
So $6=(1+\alpha_{1})(1+\alpha_{2}).....(1+\alpha_{k})$
I just don't know about the next step...

3. I tried different values of n:
Well 2^5 works
(5+1)=6
3^1*7^2
(1+1)(2+1)=2*3=6

So n=p^5 or p1^2*p2?
And 12 is the smallest n?

4. hint: since 6 has only one non-trival factorization 6=2*3, we obtain k=2, and a_1 = 1, a_2 = 2.